One should verify that x1,x2 and x3 are not integers. Here is a sketch of one way to do this. First, prove by induction that the sequence {xn} is increasing. Next, since 1<x1<2 and x4=3, it suffices to show that neither x2 nor x3 is 2. As for x2, note that x1<3/2 (since x13=3<27/8 ). Thus
x2=x1x1<(3/2)3/2=27/8<2
To show x3>2, show that x1>2 and do a similar manipulation.
Note. x2 and x3 are integers raised to radicals, e.g.,
x2=3(31/9)
There is a general theorem which says (as a special case) that such numbers cannot be integers (except when the base is 0 or 1), but the proof is very deep.